ANALYTICAL INVESTIGATION OF [Book III. It will readily appear by the same process, when all powers of the eccentricities above the second are rejected, that u' = 1{1+e'(1-te') cos (c'mv-w')+e2 cos (2c'mv-25')) (218) al+mee'cos(cv-c'mv-w+w')-mee'cos(cv+cmv-w-w')) 703. By the same substitution, cos (v - v') = cos (v-m'v) cos + + sin (v-mv) sin ψ; but hence and cos y = 1 4*+ &c. sin y = ψ mv) me cos (v + me cos (v cos (v - mv) sin (v - mv) &c.; = cos (v - mv) + me cos (2cv 43 + &c. v + mv – 25) - me cos (2cv + v - mv 20) + my cos (2gv - v + mv – 20) my cos (2gv + v - mv - 20) + mer cos (v-mv-+-2gv-cv+-5-20) + e'(1-e') cos (v-mv-c'mv+w') - e'(1-2) cos (v-mv+c'mv-') + &c. &c. (219) Thus the series expressing cos (v-v') may extend to any powers of the disturbing force and eccentricities. which shows that cos (2v - 2v') may be readily obtained from the developement of cos (v - v') by putting 2v for v, and 24 for ψ; and the same for any cosine, as cos i(v - v'). 705. Again, if 90° + v be put for v, cos (v – mv) becomes cos {(v + 90°) (1 – m)} = sin (v - mv) ; hence also the expansion of sin (v - v') may be obtained from the expression (219), and generally the developement of sin i(v – v') may be derived from that of cos i(v - v'). Thus all the quantities in the equations of the moon's motions in article 695 are determined, except the variation du, du', dv', and ds. 706. It is evident from the value of + u in equation (209), d'u that u is a function of the cosines of all the angles contained in the products of the developements of u, u', cos (v - v') cos (20 – 20') &c.; and du, being the part of u arising from the disturbing action of the sun, must be a function of the same quantities: hence if A0, A1, A2, &c. be indeterminate coefficients, it may be assumed, that adu = A. cos (2v-2mv) + A. e.cos (2v-2mv-cv+w) + A. e.cos (2v-2mv+cv-w) + A. e.cos (2v-2mv+cmv-w') + A. e.cos (2v-2mv-c'mv+w') + Ase.cos (c'mv - w') + A. ee.cos (2v-2mv-cv+cmv+w-w') + A, ee cos (2v-2mv-cv-c'mv+w+w') + Agee.cos (cv+c'mv-w-) + A, ee.cos (cv-c'mv-w+w') + A10 e. cos (2cv-20) + Ane.cos (2cv-2v+2mv-25) + Αιγ2. cos (2gv-20) + A13 y cos (2gv-2v+2mv-20) + Au el cos (2c'mv- 20') + Ais eye cos (2gv-cv-20+0) + A16 ey2 cos (2v -2mv-2gv+cv+20-5) (220) The term depending on cos (cv - 5) which arises from the disturbing action of the sun is omitted, because it has already been ineluded in the value of u. 707. It is evident from equation (210) that ds, the variation of the tangent of the latitude, can only have the form ds = Bo y sin (2v-2mv-gv + 0) + B. ex sin (gv+cυ-θ-w) + B3 ey sin (gv-cυ-θ+π) + B. ex sin (2v-2mv - gv+cv+0-0) + B1 ey sin (2v-2mv+gv-c-0+5) + B. ey sin (2v-2mv-gv-cου+0+0) + B, e'y sin (gv+c'mv-θ-') + B. e'y sin (gv-cmv-0+) + B, e'y sin (20-2mv-gv+c'mv+0-w') + B10 ey sin (20-2mv+gv-c'mv-0+w') + Bu ey sin (2cv-gv-25+0) + B12 ery sin (2v-2mv-2cv+g+20-0) + B13 ery sin (2cv+gv-2v+2mv-25-0) (221) 708. The variation in the longitude of the earth from the action of the planets troubles the motion of the moon. Equation (216), when (nt + €) is put for dv, gives δύ' =md(nt+€){1+2e' cos (c'mv-w') -e cos (2c'mv-20')}(222) But dv or 8(nt + €), arising from the disturbing force, is entirely independent of equation (213), which belongs to the elliptical motion only; and from equation (211) it appears that if Co, Cg, &c. be indeterminate coefficients, d(nt + €) = C. sin (2v-2mv) + Ce sin (2v-2mv+c'mv-w') (223) By this value, equation (222) becomes δυ' = m{C + C, e2 + C1o e'} sin (2v – 2mv) (224) 709. But the longitude of the earth is troubled by the action of the moon as well as by that of the planets, and thus the moon indirectly troubles her own motions. In the theory of the earth it is found that the action of the moon occasions the inequality in the earth's longitude, and thus the whole variation of v is dv = m { C + C, e' + C1o e2 } sin (2v – 2mv) (225) where is the ratio of the mass of the moon to the sum of the masses of the earth and moon. 710. The parallax of the moon is troubled by both these causes, but that arising from the action of the planets may be omitted at present. The moon's attraction produces the inequality dr' = pr cos (v - v') in the radius vector of the earth, and consequently the variation δυ' = in the solar parallax. du 711. Lastly, is obtained from equation (214). dv 712. Thus every quantity in the equation of article 695 are determined, and by their substitution, the co-ordinates of the moon will be obtained in her troubled orbit in functions of her true longitude. The Parallax. 713. The substitution of the given quantities in the differential equation (209) of the parallax is extremely simple, though tedious. The first term when the higher powers of so are omitted; putting 1 {1+e+r+6+y*(1+e^-12) cos (2gv-20) }. and for s2 becomes 1 h2(1+s2) a Again, u = 1 13 a/3 {1+e+ 3e' cos (c'mv - w') + &c. } 2 u3 = a3 {1 − y - 3e cos (cv - 5) + &c.}; m2 2 m'u3 2h2u3 = {1 + e2 + y2 + 3e (1 + e2 + 2) cos (cv-w) + &c.} terms, 6 is omitted, being of the fourth and, by comparing their coefficients with observation, serve for the determination of the ratio of the parallax of the sun to that of a a the moon; but as it is a very small quantity, any error would be sensible, and on that account the approximation must extend to quantities of the fifth order inclusively with regard to the angle v-v'; but in every other case, it will only be carried to quantities of the third order. 715. Attending to these circumstances, and observing that in the |